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Robust Nonlinear Adaptive Flight Control for Consistent Handling Qualities Rolf Rysdyk and Anthony J. Calise, Senior Member, IEEE
Abstract—A flight control design is presented that combines model inversion control with an online adaptive neural network (NN). The NN cancels the error due to approximate inversion. Both linear and nonlinear NNs are described. Lyapunov stability analysis leads to the online NN update laws that guarantee boundedness. The controller takes advantage of any available knowledge for system inversion, and compensates for the effects of the remaining approximations. The result is a consistency in response which is particularly relevant in human operation of some unconventional modern aircraft. A tiltrotor aircraft is capable of converting from stable and responsive fixed wing flight to sluggish and unstable hover in helicopter configuration. The control design is demonstrated to provide a tilt-rotor pilot with consistent handling qualities during conversion from fixed wing flight to hover.
This aspect is desirable to reduce the workload of a human operator of complex systems like tiltrotor aircraft. The tiltrotor aircraft is capable of converting from stable and responsive fixed wing flight to sluggish and unstable hover in helicopter configuration. It is desirable to provide the pilot with consistent handling qualities during a conversion from fixed wing flight to hover, which would typically occur during the high-workload landing phase of flight. A linear model inversion architecture is adopted by frequency separation. The architecture provides for a model following setup with guaranteed performance. A rigorous proof shows how boundedness is guaranteed.
Index Terms—Adaptive control, flight control, neural network (NN), nonlinear control.
I. INTRODUCTION
SUMMARY
T
HE use of modern technology in flight control systems allows for the design of consistent handling qualities even during radical aircraft configuration changes. The same technology can also provide a fault tolerant control system that is capable of providing consistent handling while the vehicle is damaged. Although a number of ad hoc designs have been successfully demonstrated, their implementation by more conventional means lacks robustness and is prohibitively labor intensive. Nonlinear Adaptive Control provides consistent performance that is superior to more conventional controller designs. It combines model inversion control with adaptive neural network (NN) compensation that cancels the inversion error. Both linear and nonlinear NNs are applied. Lyapunov stability analysis resembling conventional adaptive control determines the update laws. The nonlinear NN provides a more powerful application based on its universal approximation property. If an approximate model of the system is available, the controller architecture can take advantage of that information and compensates for the effects of approximation. When used in a model-following set-up, this results in consistent responses.
Manuscript received April 30, 2003; revised April 26, 2005. Manuscript received in final form May 17, 2005. Recommended by Associate Editor P. K. Menon. R. Rysdyk is with the Department of Aeronautics and Astronautics, University of Washington, Seattle, WA 98195 USA (e-mail:
[email protected]). A. J. Calise is with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail: anthony.calise@ aerospace.gatech.edu). Digital Object Identifier 10.1109/TCST.2005.854345
Next generation aircraft may differ radically from their predecessors, presenting control designers with interesting challenges and opportunities. Examples include: low-observable and supermaneuverable tailless fighter aircraft like the X-36 in Fig. 1 and [1], aircraft capable of flight in multiple configurations like the tilt-rotor described in Section II-B, and remotely piloted and autonomous vehicles unconstrained by human occupants. The desire for enhanced agility and functionality demands performance over an increased range of conditions characterized by large variations in dynamic pressure and aerodynamic phenomena. Furthermore, the use of nonlinear actuation systems increases the complexity of the control design. Alternatively, variation in response may occur due to damage or component failure, requiring rapid reconfiguration of the control system to maintain stable flight and reasonable levels of handling qualities. Therefore, there is interest in real-time direct adaptive control methods with guaranteed performance. The most widely studied approach to nonlinear control involves the use of transformation techniques and differential geometry. The approach transforms the state and/or control of the nonlinear system into a linear representation. Linear tools can then be applied in terms of a pseudocontrol signal, which is subsequently mapped into the original coordinates via inverse transformation. This broad class of techniques is most commonly known as feedback linearization (FBL) [2]. FBL theory has many applications in flight control research. Meyer and Cicolani included the concept of a nonlinear transformation in their formal structure for advanced flight control [3]. Menon et al. used a two-time-scale approach to simplify the linearizing transformations [4]. A fixed Jacobian can provide a dynamic inverse for nonlinear plants, leading to asymptotic tracking of desired trajectories, with bounded error [5]. Dynamic inversion techniques have been investigated at great length for application to super-maneuverable aircraft [6]–[8]. A drawback of dynamic
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inversion is its vulnerability to modeling errors [9]. Therefore, several techniques have been proposed to provide robustness to sources of uncertainty, which include unmodeled dynamics, parametric uncertainty, and uncertain nonlinearities [9]–[11]. Many of the results in adaptive control are derived from Lyapunov stability theory [12]. Although adaptive control has a long history, it did not gain favor until 1980, when important results guaranteeing closed-loop stability were obtained [13]. Several efforts concentrate specifically on direct adaptive control of feedback-linearizable systems [14], [15]. The sensitivity of some adaptive schemes to disturbances and unmodeled dynamics prompted investigation of robust adaptive control for linear systems. Possible tools include the use of a dead-zone to maintain bounded errors in the presence of noise [16], parameter projection techniques to provide robustness to unmodeled dynamics [17], and methods for improving robustness of adaptive nonlinear controllers using backstepping [18]. While treatment of disturbances and uncertain nonlinear functions is now common, fewer efforts address robustness to unmodeled dynamics. Some exceptions include application to high-performance aircraft [19], and use of the backstepping paradigm [20]–[22]. Artificial NNs have the ability to approximate continuous nonlinear functions [23], [24]. One advantage of the NN over simple table lookup approaches is the reduced amount of memory and computation time required. In addition, the NN can provide interpolation between training points with no additional computational effort. NNs function as nonlinear adaptive control elements and offer advantages over conventional linear parameter adaptive controllers. Survey papers commenting on the role of NN technology in flight control design have been contributed by Werbos [25] and Steinberg [26], [27]. We focus on the use of a direct adaptive NN-based control architecture that compensates for unknown nonlinearities in a feedback linearizing control framework. Previous works include applications to helicopters and tiltrotors [28]–[31]. fighter aircraft [32], [33], agile missiles, and guided munitions [34]–[36]. In a second part of this paper, we address the issue of robustness to unmodeled actuator dynamics, which is treated by modifying the adaptation law with dynamic nonlinear damping [21], [35], which to our knowledge is the first time this has been developed for fully nonlinear adaptive systems.
II. CONTROLLER ARCHITECTURE A. Approximate Inversion The objective of this paper includes demonstration of the NN capability of adapting to errors caused by using an approximate inverse model. Unmodeled dynamics originate from the linearization used for the nominal inverting controller. This includes linearization of dynamics that are nonlinear with respect to the control variables. Additionally, any cross-coupling between fast rotational states and slow translational states is neglected in the inversion. In this paper, we consider the case where the number of outputs equal the number of available control inputs. By approximating the system dynamics to be linear in states and control variables, the approximate feedback
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Fig. 1. Boeing/McDonnell Douglas X-36 tailless fighter agility research aircraft.
Fig. 2.
Adaptive NN augmented model inversion architecture.
linearizing control design process reduces to construction of a model inverting control law. This control law represents a linear, state-dependent, transformation from pseudocontrol space to control space. The conditions necessary for exact FBL of nonlinear systems are well researched [2]. Formal definitions for how “close” an approximate inverse model output is to the exactly linearized output are also known, including for model following control design [37]–[39]. A regulator can be added to the pseudocontrol to drive the error between actual output and model output to zero. Disturbances and variations in plant dynamics may be handled this way. Such methods have been successfully applied in flight control [40]. A rigorous justification for neglecting the moment-to-force coupling in aircraft dynamics in controller design is provided in an approximate input–output linearization theory in [41]. This paper is extended to tracking control for nonaffine systems in [42]. In our application, exact FBL will lead to linear behavior from pseudocontrol to vehicle-state. The use of an approximate model in the control law induces an inversion error. We consider here a definition of the inversion error as in [32]. Fig. 2 contains a diagram of the controller architecture. Consider the aircraft dynamics represented as (1) Let represent an exact FBL control law. That is, the transfor, given by mation (2) The transformation from pseudocontrol to state then becomes (3) Instead of control law (2), we apply an approximate transformation (4)
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where notation is used to distinguish these plant trajectories from those resulting from applying resulting from using . If and are close, e.g., in the sense of [37] and [38], and will be close, and a regulator can be added then . Using the approximate inversion to to bring control law (4), in light of these results, allows us to express the aircraft dynamics as follows: (5) where
is the inversion error defined as (6)
and , with representing the action of an adap. The closeness of the approxitive NN designed to cancel mation is captured by the inversion error, which we may express in terms of the pseudocontrol signal as (7) The inversion error depends on , whereas will be . This poses a fixed-point problem with designed to cancel existence and uniqueness of its solution guaranteed with the following assumption: is a contraction Assumption 1: The mapping over the entire input domain. This implies
which can be stated as (8)
Fig. 3.
XV-15 tilt-rotor in helicopter configuration.
[43]. As outside visual cues degrade and flight-path precision requirements increase, as for example with civilian instrument meteorological conditions (IMC), the need arises for attitude stabilization and even attitude control for precise hovering and low-speed flight. As speed increases, the need for more roll maneuverability emerges, leading to a relaxation of roll control to a rate response type. Similarly, the desired control in yaw axis changes from heading command to yaw rate control with turn coordination (TC). The considerations involved in a tiltrotor IMC approach procedure, include [44]: a conversion schedule of nacelle angle with speed, from cruise to helicopter configuration in the approach, and vice versa for the missed approach procedure; deployment or retraction of flaps depending on nacelle angle, speed, and glide-slope; switching between control augmentation types, and; desired altitude and speed trajectories. Consider the aircraft rotational dynamics represented by [45] and [46] (9)
Expression (8) implies the following two conditions: 1) ; 2) B. Tiltrotor Application Tiltrotor aircraft combine the hover performance and control of a helicopter with the cruise speed and efficiency of a turboprop airplane. Tiltrotor aircraft feature wing-tip mounted prop-rotors that can be rotated from a vertical orientation for takeoff and landing to a horizontal position for efficient fixedwing-borne flight for high-speed cruise, Fig. 3. There is an interest in large tiltrotor transports, which promise to relieve airport congestion by replacing commuter aircraft and freeing up runway slots. The flight mechanics of a tiltrotor present both opportunities and challenges to the control designer. Prop-rotor movement from the vertical position in helicopter mode, toward the horizontal airplane mode position, rapidly accelerates the aircraft while orienting prop-rotor thrust to its optimum position. Conversely, up, and aft movement of the prop-rotors, required to prepare for a vertical landing, provides the drag needed to decelerate but at the same time produces undesirable additional lift, which the pilot must counteract with appropriate flight-path control. Two common types of stability and control augmentation systems (SCAS) for aircraft are referred to as rate command attitude hold (RCAH) and attitude command attitude hold (ACAH)
which are, respectively, the where Cartesian components of velocity along the body-fixed axes and contains the angular rates about the Euler angles, the body-fixed axes, and is the control input, respectively, referred to as the lateral cyclic, longitudinal cyclic, rudder, and collective. The approximate model is based on dynamics linearized about a nominal operating point, with the rotor dynamics residualized (10) where , and , respectively, represent the aerodynamic stability and control derivatives in the usual Jacobian sense. The collective/throttle control position is treated as one of the rela. The intively slow translational states, puts of interest are the controls of moments about the body axes, . The inverting control law is constructed from (10) by replacing the angular accelerations with their desired values and solving for the control perturbations. This results in the following control law: (11) where the hats indicate that we may allow some further uncertainty in approximations of , and . The inversion error is (12)
RYSDYK AND CALISE: ROBUST NONLINEAR ADAPTIVE FLIGHT CONTROL
The effect of as
can be represented about the body-fixed axes (13)
The components of are related to those of the pseudocontrol as explained next. The pseudocontrol for the three rotational degrees of freedom is designed in terms of body angular rates as (14) where is the output of a NN, and is the output of a linear controller operating on a tracking error signal. A variety of linear control designs can be used to produce . We use a combination of command filter and classic proportional–integral derivative (PID) control to provide the model following setup indicated in Fig. 2. If the poles of the command filter are co-located with those of the PID control, then in case of ideal inversion the architecture reduces to conventional explicit model following. With the approximate inversion, the PID control design affects the NN performance and, thus, determines is dethe tracking error transient. The pseudocontrol signal signed as follows: (15)
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approach-to-landing stages of flight benefits from attitude-command in the longitudinal channel. The roll and yaw channels are commonly designed for rate-command. The components of the for this combination of augmentation desired acceleration are related to the components of the pseudocontrol as follows: (19) (20) By using the derivative with respect to time of the pitch kinematic expressions, the desired angular acceleration about the body axis can be solved for where
(21) (22)
where is shorthand for , etc. Thus, the desired pitch rate, given the commanded attitude and yaw rate signals, is now seen to be (23) To see the effect of this construction on the inversion error, nocan be tice that by combination of (11), (12), and (23), represented as a function of the states and the pseudocontrol. In . We may reppitch channel, the error is a function of and in the Euler pitch-attitude dynamics as resent the effect of (24)
with (16) where the signals and their derivatives are the outputs from command filters, and for each component the tilde represents a command tracking error (17) The command filters are used to specify the rotorcraft handling qualities [43]. In this paper, the signals and will be designed to provide RCAH in roll and yaw, and will provide ACAH in pitch. The integral action is added in the roll and yaw channels to provide the attitude hold in those channels. Integration and adaptation wind-up can be prevented with pseudocontrol hedging (PCH) [47]. Handling qualities specifications and actuator performance allow the tracking error transient to be fast relative to the dynamics of the command filter, while maintaining bandwidth separation from actuator dynamics. The following assumption can therefore be satisfied by design. Assumption 2: Let the external commanded input and its first and second derivatives be bounded, for example such that (18) Similar assumptions are made for the roll and yaw channels. In what follows, a full three-axes control augmentation will be presented and demonstrated, while the pitch channel will be used as a detailed example for design and analysis. For the tiltrotor application, our focus is on the approach to landing phase of flight. The high workload associated with conversion from fixed-wing to helicopter flight combined with approach procedures calls for attitude-hold in all channels. The need for precision during the
where represents the th component of vector Combining (14), (15), and (24), we obtain
. (25)
where is the pitch component of the inversion error when represented in the Euler frame. The left-hand side of (25) represents the tracking-error dynamics. The right-hand side is the network compensation error, which acts as a forcing function on the tracking error dynamics. With completely canperfect NN performance, the NN output . cels III. NEURAL NETWORK STRUCTURE A. Linear NN Structure An online NN is defined by its structure and its update laws. A linear NN structure consists of any linearly parameterized feedforward network that is capable of approximately reconstructing the inversion error. Reference [48] uses radial basis functions (RBFs) because these functions are universal approximators even when the network is linearly parameterized. However, it is well known that RBFs are poor at interpolation between their design centers, and a large number of such basis functions are needed for networks with multidimensional input vectors. In [32], RBFs were used to capture variations in Mach number, because in the trans-sonic region, these variations are difficult to represent by polynomial functions. In the current implementation, a single-layer sigma-pi network is used. The inputs to the network consist of the state variables, the pseudocontrol and a bias term. Fig. 4 shows a general depiction of a
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Fig. 5. Structure of a SHL perceptron network.
where Fig. 4. Linear in the parameters Sigma-Pi NN structure designed for ACAH in the pitch channel.
sigma-pi network. The values represent the weights associated with a nested kronecker product of input signal categories, are the and therefore they are (binary) constants. The values variable network weights. The input–output map of the linear NN is represented as (26) Here is the NN output, and are NN input, and the bar indicates possible normalization. The vector consists of selected normalized elements of the plant state and a bias term. The vector of basis functions is akin to the regressor-vector in adaptive control texts. The basis functions are made up from a sufficiently rich set of functions so that the inversion error can be accurately reconstructed at the network output. The basis functions were constructed by grouping the normalized inputs into three categories. The first category is used to model inversion , since error due to changes in airspeed the (dimensional) stability and control derivatives are strongly dependent on dynamic pressure [45]. In allowing the plant to be nonlinear and uncertain in the control as well as in the states, the inversion error is a function of both state and control signals, and these are therefore contained in the second category. Furthermore, for error compensation in the pitch channel, both and should be input. The third category is used to approximate higher order effects due to changes in pitch attitude. These are mainly due to the transformation between the body frame and the inertial frame
The Kronecker product results in a combination of polynomial , and cross terms. The bias value signals that will include in each class is normalized at 0.1, allowing for a pure bias comof multiplied by . ponent in B. Nonlinear NN Structure Consider again the architecture in Fig. 2. We now replace the linear-in-the-parameters NN with a single-hidden layer (SHL) “perceptron” NN. NNs with a SHL structure are more powerful than the linear NNs because they are universal approximators [23], [24]. Although the controller architecture does not reflect many changes from the linear NN application, there are differences in the stability analysis. Most of the added complexity can be traced back to the backpropagation update laws of the SHL-NN, and its associated Taylor-series approximation [49]. Fig. 5 shows the structure of a SHL perceptron NN. The input–output map of a SHL network can be represented as (27) where
and (28)
Here , and are, respectively, the number of input and outputs, and number of hidden layer neurons. The scalar funcis a sigmoidal activation function that represents the tion ‘firing’-characteristics of the neuron. (29) The factor is known as the activation potential. For convenience define the two weight matrices
Finally, the vector of basis functions is composed of combinations of the elements of , and by means of the Kronecker product
.. .
..
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and
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and are the estimates of the ideal parameters. where We need to make the following assumption. Define .. .
..
(36) .
It is also convenient to define a vector
allows for the thresholds where the weight matrix . Define
and let imply the Frobenius norm. Assumption 3: The norm of the ideal NN weights is bounded by a known positive value
as (30)
(37)
to be included in
Let and be the estimates of, respectively, and . and define the hidden-layer Define output approximation error as
(31) is an input bias that allows for the thresholds to be included in the weight matrix . With the previous definitions, the input–output map of a SHL Perceptron can be written in matrix form as
(38) To backpropagate the estimation error through the NN hiddenlayer, we use a Taylor series expansion about the current esti, where we are mate of the hidden layer output, specifically interested in
(32) The representation of the NN output given in (32), may be used to represent a linear-in-the-parameters NN by specifying , and constructing by using well distributed radial basisfunctions [48], or by providing polynomial combinations of the elements of [28]. IV. INVERSION ERROR COMPENSATION Consider a NN approximation of an inversion error (33) , where an upperbound defined in what where follows. The vector is referred to as the NN reconstruction error, or residual error. The vector provides the set of basis . We may functions that serves to approximate the function include adaptive parameters in this set, as is the case with perceptron NNs, resulting in nonlinearly parameterized NN output. The NN input is made up of selected elements of the state vector and pseudocontrol. The selection of the elements of is done through careful assessment of the inversion error [28]. , and are matrices of constant, not necessarily unique, pa. These parameters are ideal, rameter values that minimize of , they bring for example in the sense that, in a domain to within a -neighborhood of the error the term , where is bounded by (34)
(39) where
.. .
.. .
.. .
(40)
. and Remark 1: As an example, consider the second-order error dynamics in the pitch channel (25), which may be represented as (41) where
and
, and with (42)
Remark 2: Due to nonaffinity of the plant with respect to , is in general a function of the control, the inversion error, pseudocontrol, which includes the output of the NN. Since the output of the NN is providing compensation for the inversion error, a fixed point problem occurs. To insure that Assumption 1 holds, is input to the NN through a squashing function.
Thus, and may be defined to be values of and that minimize over . The online NN output may be represented as
A. Using the Nonlinear NN for Inversion Error Compensation
(35)
(43)
If the nonlinear NN is used, the signal
is designed as
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where is a term that robustifies against the effects of higher order terms in the Taylor series approximation in (39)
Cauchy–Schwarz inequality and the compatibility of the Frobenius norm with the vector 2-norm, it is clear that is bounded if is
(44) where a known positive constant defined later, and such that . The update law of the NN weights is designed as (45)
(51) A bounded also implies a bound on the 2-norm of , allowing concentration on the boundedness of the scalar [2]. Lemma 1: Let be constructed as (47), and suppose then
(46) where 0, and a known positive conis known stant defined later. The damping term as e-modification. In our design, with 0. The elements of represent the sensitivity of the . hidden layer to its input, for the nonlinear NN The scalar is the filtered error term
(52) and (53) A proof is given in the Appendix. From Lemma 1, we can obtain (54)
(47) with for the second-order example in the pitch channel D. Guaranteed Boundedness
The NN input design is discussed in the Appendix. The NN input can be upper bounded in terms of the tracking performance by B. Using the Linear NN for Inversion Error Compensation , i.e., If a linear-in-the-parameters NN is used with linear in the adaptive parameters , then no update of V is 0, and therefore 0, and desired,
where
0 are known. From (39)
(48) The adaptation law is given, with
0, by (49)
Remark 3: The example provides for ACAH in the longitudinal channel, i.e., a design with second-order tracking error dynamics. This design may be generalized with (50) and a size- vector, and an -matrix 0, with in canonical form that solves similar to (42), expanded to order . Remark 4: Equations (45), (46), (44), and (49) are stated for a single channel setup, i.e., with one NN output. The statements may be generalized for MIMO implementation. In fact, the strength of the architecture lies in the cancellation of a nonlinear and possibly multidimensional inversion error which may include coupling of multiple states and control effects. A MIMO application that takes advantage of this capability is detailed in [50]. C. Filtered Error Bound imply the two-norm in case of vectors and the Let Frobenius norm in case of matrices. The construction of in (47) can be seen as an error filter, see the Appendix. From the
where . With these results the higher order terms associated with the back propagation are bounded from above by (55) 0, known. Let be defined as the NN approxiwhere mation error plus the higher order effects of back propagation through the nonlinear NN:
Considering the properties of the NN structure, (55), and (34), an upper bound on in terms of is
Combining this with (A-4), then (56) where 0 are known. Remark 5: Using the facts that and it is possible to find a known upper bound
such that (57)
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Fig. 7. Geometric representation of the effect of commands. Fig. 6.
r (Z ) on the allowable
Geometric representation of sets in the theorem.
In the formulation of this control problem, the error space can be considered as consisting of a subspace associated with tracking, and one associated with the NN weights. In the tracking error subspace, let (58) and represent, respectively, the maximum and minLet be deimum eigenvalue of positive–definite matrix . Let fined by, Fig. 6 (59) where (60) Similarly, in the NN weight subspace with learning rate where the identity matrix of appropriate dimension, let
, (61)
and . Theorem 1: If of is sufficiently large, such that main
and if the do, with
signals can be pictured in a geometric representation of the intersection of the sets with the -subspace, Fig. 7. This shows that commands of larger magnitude imply smaller values for . This may be interpreted to mean that to limit the closed-loop bandwidth, smaller NN learning rates may be required when allowing for more aggressive command tracking.
V. TILT-ROTOR SIMULATION RESULTS The aircraft is simulated using the real time flight simulation model of a tilt-rotor aircraft Generic Tilt-Rotor Simulator (GTRS) developed at the NASA Ames Research Center in support of the XV-15 and V-22 programs [51]. The code was extended to include actuator dynamics and nonlinearities. The following results summarize the controller performance in all channels. In the designs for human piloting investigated here, both the linear and SHL NN performed similarly, with only minor differences due to NN sizing and learning rate. It should be noted however that in more aggressive and highly nonlinear applications, the SHL NN has been demonstrated to have superior performance [1], [36], [52]. A. Command Filter Design
(62) and similarly, the domain with
of
sufficiently large such that
(63) where , then all the signals in a system defined by (41) and (49) will remain bounded. A proof is presented in the Appendix. Remark 6: The term represents the effects of back propagation, which is not applicable to the linear NN. For the linear , therefore NN, (64) Remark 7: The external command signals were assumed to be bounded, as in Assumption 2. The effect of these command
The GTRS code includes the existing XV-15 control augmentation, here referred to as original SCAS. This SCAS is gain scheduled with speed and with mast-angle. In the longitudinal channel, it provides ACAH and RCAH, depending on the mode selected by the pilot. The ACAH setting was used for the comparison in the following results. The linear PID controller dynamics in (15) were designed so that the NN error settles within 0.5 s. If the model following performance in the architecture of Fig. 2 is guaranteed over the frequencies of interest for handling qualities (0.2–6.0 rad/s), the Airworthiness Design Standard ADS-33D requirements [43] can be implemented by means of, respectively, a second- and first-order command filter. In that case, the Lyapunov stability offers a guarantee of the ADS-33 performance. Attitude Command can then be implemented with a second-order pitch-angle command filter, such that its dominant complex poles provide minimal overshoot and a 5% settling time of 1.5 s. This provides Level 1 handling characteris-
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Fig. 8. Longitudinal SCAS comparison. The aircraft starts out trimmed at nominal-operating configuration. During the first 30 s it responds using the original gain scheduled SCAS. The aircraft remains within 10 Kts and within 32 the model inversion SCAS is activated as 250 ft of its altitude. At t evidenced by the NN weight histories.
=
tics in the pitch channel. Level 1 Rate Command handling qual2 rad/s. With the ities in roll has a phase bandwidth BW roll-yaw coupling small, the time constant of the roll response is approximately the inverse of the bandwidth, therefore (65) This provides the roll command filter with a 5% settling time of approximately 1.5 s. The setup for the yaw-channel is similar 0.25 s. These command filter designs with a time constant, will provide Level 1 handling qualities when the augmented aircraft can indeed follow the command filter dynamics up to these frequencies. B. Control Augmentation for the Longitudinal Channel A comparison of the original SCAS with the NN augmented model inversion augmentation is presented in Fig. 8. The improvement in pitch response is clearly visible. The same performance was obtained at various points in the operation envelope and at various configurations, including at 35 000 ft in aircraft configuration. The augmentation in the lateral channel is able to provide similar performance and consistency in response. To further evaluate the performance of the NN adaptive control, a pilot model was developed [53]. The pilot model is able to follow desired altitude, speed, and conversion profiles, Fig. 9. Note that the NN weight histories return toward nominal as the tiltrotor is maneuvered back to trim. C. Control Augmentation for the Lateral Channels Two common modes of augmentation in the yaw channel are heading hold (HH) and TC. In a tiltrotor aircraft, a transition between these modes is typically made between 25 and 45 Kts. HH is inherent to the controller architecture as attitude stabilization in yaw. TC pertains to the need for coupling between the roll and yaw motion to provide for a comfortable and efficient change in heading. Etkin [45] defines a coordinated turn as a turn wherein the rate of change in heading is constant, and the resultant of gravity and centrifugal force at the center of mass lies in the
Fig. 9. Simulated approach to minimum descent altitude, starting from 1000 ft in nominal operating configuration at 30 Kts helicopter configuration. The desired descent rate is 1000 fpm. The primary control for establishing the descent rate is the collective. This has an effect on the velocity, which is subsequently controlled by the pilot model through pitch commands. Note that the NN weight histories return toward nominal as the tiltrotor is maneuvered back to trim.
plane of symmetry. This definition approximates closely a piloted flight on the turn-and-bank indicator. For the implementation of TC, only the command filter dynamics are considered, i.e., we assume perfect tracking of yaw rate command. The desired yaw rate can then be expressed as (66) is the gain of the tracking dynamics in the where roll-channel, and is the acceleration along the bodyaxis. The dynamics due to are neglected as its value is usually not available from measurement. For the guidance outerloop we take the yaw rate to respond as directed by a first-order command filter. Grouping the dynamics with the difference between measured and actual effects of roll-rate and gravity, as (67) Equation (67) indicates that the effects of gravity and roll rate are washed out. The effects due to are not, but if 0 and with mild maneuvering, this augmentation will keep suppressed. Closing the loop results in an expression for the lateral-acceleration transients (68) Fig. 10 compares the performance of the original XV-15 SCAS with the model inversion controller. The original augmentation provides rate command and attitude command with attitude retention (ACAH) upon selection. No HH or automatic TC is available for comparison. Figs. 11, and 12 demonstrate the augmented response in the lateral channels given. The controller architecture inherently provides for HH. Fig. 13 displays the difference in performance between TC and HH mode. HH would typically be used at speeds below 45 Kts,
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Fig. 10. Lateral SCAS comparison; response to a square wave lateral stick 15 s, at that input at hover, at 5000 ft. The original SCAS is active until t moment the model inversion SCAS is switched on. The 2-s fluctuation in roll rate is due to the coupling of the yaw augmentation that is now providing HH.
=
Fig. 11. Roll channel command and response to roll rate doublet, with automatic TC, leading to a 45 banked turn. The tiltrotor is flying 170 Kts, at 10 000 ft, in cruise configuration. Automatic TC is implemented as in (66), with a 0g .
=
TC is used in cruise flight. The 30 Kts flight is representative for HH operation, but rather an extreme for the TC mode. The controller is able to provide good performance. D. Handling Qualities ADS-33D [43] requirements have a specific focus, in each channel on: small amplitude (high frequency) responses; moderate and large amplitude responses; and response to disturbance (attitude hold performance) [31], [54]. The simulation model includes XV-15 actuator dynamics and nonlinearities. However, the investigated maneuvers did not lead to actuator saturation. The small amplitude response to control inputs is tested through rad/s. The unaugmented XV-15 a frequency sweep, displays conventional first-order roll rate characteristics where aileron deflections produce a proportional roll rate at low frequencies, and acceleration at high frequencies. The dominant time constant is approximately 1 s. The critical bandwidth is determined by the 45 phase margin [43]. The effective phase time
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Fig. 12. Yaw channel response, control surface deflections and TC performance in roll rate doublet of Fig. 11.
Fig. 13. HH versus TC in helicopter configuration, 30 Kts at 3000 ft. Given a doublet roll rate command, the performance in roll channel shows no differences. The maximum bank angle is approximately 25 . In HH mode, the heading remains within 0.25 of its trim value, and the aircraft reaches a lateral velocity of 20 Kts.
delay is negligible since the phase curve is nearly flat where it is near 180 . The bandwidth-phase-delay requirement, for Level 1 roll response, is similar to the one in pitch response, referred to as Small Amplitude Roll Attitude Changes. It is a requirement for 2 rad/s. If the roll-yaw coupling a phase bandwidth of is small then the time constant of the roll response is approxi. mately the inverse of the phase bandwidth, 0.5 s So a command filter design with a time constant should result in Level 1 handling qualities. Fig. 14 shows the results of the frequency sweep. Moderate amplitude changes (attitude quickness) requirements and Level 1 requirements for so-called large amplitude roll attitude changes for IMC operations were also achieved [54]. Figs. 15, and 16 give the pitch response to a longitudinal square wave stick input. The XV-15 is flying at 2000’, at approximately 100 Kts, in helicopter configuration. The figures contrasts the commanded pitch angle with the response using model inversion only, using the linear NN, and using the SHL
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Fig. 14. Roll attitude stick transfer function frequency response at cruise flight. The magnitude of the stick deflection covers 10% of total travel, representative under these conditions, compared with a Bode-plot of the command filter dynamics. The results from the spectrum analysis match the command filter for all frequencies of interest, and the augmentation satisfies the Level 1 bandwidth requirements.
Fig. 16. Inversion error and NN compensation, and actuator activity. The NN 10. The SHL weights are initialized with zero weight. The linear NN has NN robustifying gains were kept relatively small in this demonstration, K = 1, and = 100 and = 100. These values K K 1.0 and Z provide for a similar response as the linear NN compensation. Though in this demonstration the different NNs performed similarly, other applications have shown the advantages of the more powerful nonlinear NNs [32], [33].
=
=
=
=
research has shown utility in the areas of cost reduction and improved flight safety. An important improvement is robustness with respect to actuator dynamics and limits. The adaptive laws may be extended to provide for robustness with respect to unmodeled dynamics [55]. PCH is a powerful extension which may be used to shield the NN adaptation from specific dynamics or actuator limits [47], [56]. PCH allows adaptation-while-saturated, which provides smooth operation when un-saturating. A draw-back of nonlinear adaptive control as presented here is the need for full state feedback. The development of an output feedback formulation is a major extension to the presented work [57], [58]. APPENDIX I NETWORK INPUT DESIGN A. Analysis of the Inversion Error Fig. 15. Aircraft states and tracking performance at 100 Kts helicopter configuration. The aircraft control is “released” at t 2 s. The model inversion architecture is activated at t 3 s.
=
=
NN. The manuever provides for a demonstration of operation away from the nominal operating point, visible as the tracking error, and in terms of inversion error in Fig. 16. This figure includes the NN output, which compensates for the inversion error (dotted line), as indicated in (25).
Similar to (12), concentrating on ACAH in the longitudinal channel, we consider (A-1) We use a SHL NN to compensate for
With the model used for inversion given by VI. CONCLUSION AND FURTHER WORK A controller architecture, which combines adaptive feedforward NNs with FBL, has been outlined and its effectivenes demonstrated on a tilt rotor aircraft. The boundedness of tracking error and control signals is guaranteed. The architecture can accommodate both linear-in-the-parameters NNs, as well as SHL perceptron networks. Theoretical and experimental
the functional dependency of
is
(A-2)
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To approximate the inversion error, the NN input should contain all the necessary information. Notice from (A-2) and (44) that
APPENDIX III PROOF OF THEOREM 1 The tracking error dynamics are given by (C-1)
Therefore, the NN input is chosen as Consider the candidate Lyapunov function (A-3) where
(C-2)
is the bias value introduced in (31). For 0 and 0 the solution of
B. Bound on NN Input
is Hurwitz, and for all
Using (C-3) is
the NN output may bounded by
where 0. The design as in (A-3), Assumption 2, and Lemma 1 imply (A-4) where
0. In the sequel we will consider three cases; we start with a and let the Lyapunov analysis linear NN structure where lead to an idealized update law. Next we introduce the use of an e-modification to the update law which prevents parameter drift in the absence of persistency of excitation (PE), and finally apply a similar approach to the nonlinearly parameterized NN. Case 1: Differentiating (C-2), substituting (33), and using (C-3), gives
0, known. (C-4) APPENDIX II PROOF OF LEMMA 1
where the scalar was defined in (47). With , (C-4) is reduced to
and, thus,
Suppose , to see how a bounded implies bounded elements of , consider (47) as a first-order filter with input and output with feedback gain . Then
(C-5) For convenience of analysis choose
. Observe that
from which we may conclude where signifies the maximum eigenvalue of positive–definite matrix and
(B-1) (C-6) Also observe that result (B-1)
and, thus, using With these observations, we may write (B-2)
Expressions (B-1) and (B-2) allow us to transfer bounds from the signal , to the elements of the second-order system (41). Stability of the signal implies stability in tracking. Specifically, we can now express the bound on in terms of , expression (54).
(C-7) This result suggests an update law (C-8)
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With this update law,
is strictly negative when
This is negative either when (C-9)
Thus, , as defined in (59), is a positively invariant set of (C-1). Furthermore, define
(C-14) or when (C-15) which follows from inequality (C-6) as:
If
, this requires that
Then the minimum size of
can be quantified by (C-10)
where must be sufficiently large, so that . This is remains bounded. Furthermore, if sufficient to show that the NN output layer is persistently exciting (PE) and the aforementioned assumptions are satisfied, this implies boundedness [59]. If , i.e., no approximation error, then reof duces to the origin and 0. In this idealized case, the fact that the NN is linear in the parameters allows use of the established adaptive control methods [60] to show boundedness of , without requirement for the output layer to be PE [59]. Case 2: Consider once more (C-5). We now consider the update law of (C-8) augmented with an e-modification, and al. PE is very hard to lowing for a pretrained weight matrix provide for all but the most simple NNs. The e-modification corrects the potential parameter drift that may occur in the absence of PE
Then for , the minimum size of (64). Case 3: Consider a Lyapunov function
is quantified by
By substituting the update laws (45) and (46) and using previously introduced definition for , the derivative with respect to time of this expression may be expressed as
Similarly to (C-12)
Therefore
(C-11) The choice of in
and substitution of this update law results
Use the known bound
Substitution of the bound on
, (56), and reordering
and note that With
and further reordering
(C-12) Therefore This can be written as a quadratic form
which may be rearranged as a quadratic form
where and with when either (C-13)
, and , . The Lyapunov function derivative is negative
(C-16)
RYSDYK AND CALISE: ROBUST NONLINEAR ADAPTIVE FLIGHT CONTROL
or (C-17) which leads to expressions (62) and (63), respectively.
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Rolf Rysdyk studied at the Delft University of Technology, Delft, the Netherlands, working with the Delft research flight simulator for proprioception and motion cues experiments. He is currently an Assistant Professor in the Department of Aeronautics and Astronautics, University of Washington, Seattle. After graduation, he moved to the United States and performed duties as a commercial pilot. Subsequently, he enrolled at the Georgia Institute of Technology, Atlanta, where Dr. A. J. Calise asked him to demonstrate nonlinear adaptive control for consistent handling qualities on a tiltrotor aircraft, which formed the focus of his Ph.D. dissertation research, leading to theoretical developments that were motivated by real world control design challenges from Industry and Government. This work included application to propulsion only control of transport aircraft, with a priori methods to deal with actuator saturation. More recently, he focused on the control of multiple UAVs in a highly dynamic environment; specifically, combined task and path planning and mission tactical maneuvers for payload optimization.
Anthony J. Calise (S’63–M’74–SM’04) was a Professor of Mechanical Engineering at Drexel University, Philadelphia, PA, for eight years prior to joining the faculty at the Georgia Institute of Technology, Atlanta. He also worked for ten years in industry for the Raytheon Missile Systems Division and Dynamics Research Corporation, where he was involved with analysis and design of inertial navigation systems, optimal missile guidance, and aircraft flight path optimization. Since leaving industry, he has worked continuously as a consultant for 19 years. He is the author of over 150 technical reports and papers. The subject areas that he has published in include optimal control theory, aircraft flight control, optimal guidance of aerospace vehicles, adaptive control using neural networks, robust linear control, and control of flexible structures. In the area of adaptive control, he has developed a novel combination for employing neural-network-based control in combination with feedback linearization. Applications include flight control of fighter aircraft, helicopters, and missile autopilot design. He is a former Associate Editor for the Journal of Guidance, Control, and Dynamics. Dr. Calise was the recipient of the USAF Systems Command Technical Achievement Award, and the AIAA Mechanics and Control of Flight Award. He is a Fellow of the AIAA and an Associate Editor for the IEEE Control Systems Magazine.
